Existence of least energy sign-changing solution for the nonlinear Schrödinger system with two types of nonlocal terms

Abstract

In the paper, we are concerned with the system of Kirchhoff-Schrodinger-Poisson system under certain assumptions on $V_{1}$ , $V_{2}$ , K and f. We are interested in the existence of least energy sign-changing solutions to the system on $\mathbb{R}^{N}$ . Because two kinds of nonlocal terms $\phi_{u}$ and $\int_{\mathbb{R}^{N}}|\nabla u|^{2}$ are involved in the system, the methods are different from the Kirchhoff or the Schrodinger-Poisson system. The two nonlocal terms $\int_{\mathbb{R}^{N}}|\nabla u|^{2}$ and $\phi_{u}$ make that the functional $J(u^{+}+u^{-})\neq J(u^{+})+J(u^{-})$ . Moreover, the nonlocal term $\phi_{u}$ does not have the convergence property because of the assumption $V_{2}$ . In addition, the convergence of these two nonlocal terms are different. In the present paper, we unify the increasing property conditions on sign-changing solution in previous papers. We construct a new homotopy operator and then weaken the assumption that f is $C^{1}$ to that of f being only continuous. We prove that the system has a sign-changing solution via a constraint variational method combining with Brouwer’s degree theory.